Axiom ax-mulass 7835

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7793. Proofs should normally use mulass 7863 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7730	. . . 4 class ℂ
3	1, 2	wcel 2128	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2128	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2128	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 963	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7737	. . . . 5 class ·
10	1, 4, 9	co 5824	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5824	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5824	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5824	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1335	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  7863